Soliton resolution conjecture for the radial Yang–Mills toy model

Consider the radial equation for a smooth finite-energy solution $W(t,r)$ on the exterior region $r\geq 1$, with boundary condition $W(t,1)=0$. Define

$$Q_\lambda(r)=\frac{r^2-\lambda^2}{r^2+\lambda^2},$$

and write $Q=Q_1$. An alternating chain consists of rescaled kinks and antikinks with scales separated by $\lambda_j(t)/\lambda_{j+1}(t)\to 0$.

Soliton resolution conjecture. Any smooth, finite-energy solution $W(t,r)$ tends as $t\to\infty$, modulo sign, either to the half-kink or to a rescaled half-kink together with an alternating chain of $N$ rescaled kinks and antikinks:

$$\begin{cases} Q(r) & \text{if }N=0,\\ -Q_{\mu(t)}(r)+\displaystyle\sum_{j=1}^N(-1)^{j+1}Q_{\lambda_j(t)}(r)+1 & \text{if }N\text{ is odd},\\ Q_{\mu(t)}(r)+\displaystyle\sum_{j=1}^N(-1)^jQ_{\lambda_j(t)}(r) & \text{if }N\geq 2\text{ is even}. \end{cases}$$

Here the $\lambda_j(t)$ are continuous positive functions satisfying, for $j=1,\ldots,N$,

$$\lambda_j(t)\to\infty,\qquad \frac{\lambda_j(t)}{\lambda_{j+1}(t)}\to 0\quad\text{as }t\to\infty,$$

with $\lambda_{N+1}(t)=t$. The function $\mu(t)$ is determined by $W(t,1)=0$, which implies $\mu(t)\to 1$ as $t\to\infty$.

This conjecture proposes a complete asymptotic description by a stationary half-kink or a multi-soliton configuration with strongly separated scales. The supplied text does not state a resolution result for this exterior-domain toy model, so its status remains open.